APPROXIMATE CONTROLLABILITY OF SECOND ORDER IMPULSIVE FUNCTIONAL DIFFERENTIAL SYSTEM WITH INFINITE DELAY IN BANACH SPACES∗

This paper studies the approximate controllability of second order impulsive functional differential system with infinite delay in Banach spaces. Sufficient conditions are formulated and proved for the approximate controllability of such system under the assumption that the associated linear part of system is approximately controllable. The results are obtained by using strongly continuous cosine families of operators and the contraction mapping principle. An example is given to illustrate the obtained theory.


Introduction
Second order abstract differential systems arise in many fields such as mathematical physics and engineering, and have been extensively studied during the past few decades. The problem of controllability for second order differential systems in Banach spaces has received considerable attention recently. Kang et al. [16] studied the exact controllability for the second order differential inclusion in Banach spaces. With the help of a fixed point theorem for condensing maps due to Martelli, the authors found a control u(·) in L 2 (J, U ) such that the solution satisfies x(b) = x 1 and x (b) = y 1 . Their results depend on the following two conditions: Balachandran and Kim [3] corrected an error of the control function in [16] and pointed out that the compactness assumption of the sine family S(t) and the conditions on G 1 imply that the operator G 1 is compact and surjective, and thus by the application of Baire Category theorem X is finite dimensional [26].
Chang and Li [5] investigated the exact controllability problem for a class of second order differential and integro-differential inclusions in Banach space. They defined a control function which was similar to that in [16] and established exact controllability results. Their results also require G 1 to be invertible and its inverse to be bounded without imposing a compactness condition on the sine family S(t). However, in most of real control systems the operators S(t) are compact for t ∈ R [14, Theorem 3.2], thus their assumptions restricted the state space to finite dimensional, and the examples recovered from the abstract theory only pertain to ordinary differential equations. Recently, Henríquez and Cuevas [12] studied the approximate controllability of control systems with state and control in Banach spaces and described by a second order semilinear abstract differential equation. They compared the approximate controllability of the system with the approximate controllability of an associated discrete system. The main assumption in [12] is the approximate controllability of the corresponding linear system. To illustrate the proposed result, they applied the theory to a wave equation.
In real systems, signal processing may introduce delays. The approximate controllability results for second order semilinear abstract functional differential equations with infinite delay was shown in [13] under the assumption that the corresponding linear system is approximate controllable. In recent years, the study of impulsive second order control systems has received increasing interest, since dynamical systems with impulsive effects have numerous applications to problems arising in information sciences, electronics, biology, ecology, etc. Sakthivel et al. [21] studied the exact controllability of second order nonlinear impulsive differential systems by using a fixed point analysis approach. Moreover, Sakthivel et al. [22] obtained the approximate controllability results for second order stochastic differential equations with impulsive effects under the assumption that the associated linear system is approximately controllable. Unfortunately, in these two papers, the authors didn't consider the damped term x (·) in defining the exact and approximate controllability of the corresponding systems, which violate the controllability definition "the state variable steers some initial position to final one" because x (t) is a state variable for a second order system. Motivated by [3], Radhakrishnan and Balachandran [19] discussed the exact controllability of second order neutral integro-differential equations with impulsive conditions in Banach spaces. For the same reason as described above for [5], the results in those papers are only applicable to ordinary differential equations.
More recently, Arthi and Balachandran [2] investigated the exact controllability of second order impulsive evolution systems with infinite delay. However, they only considered x(t) without taking into account the damped term x (t) in defining the exact controllability of the second order abstract system. Up to now, to the authors' knowledge, controllability of such systems with proper definition has not been studied.
The concept of exact controllability is usually too strong and has limited applicability. Approximate controllable systems are more prevalent, and very often the results are adequate in application. Therefore, it is necessary and important to consider approximate controllability for second order impulsive functional differ-ential systems with infinite delay. In this paper, we derive results on approximate controllability of second order impulsive functional differential system with infinite delay by assuming that nonlinear function and impulses satisfy some inequality conditions, and the corresponding linear system is approximate controllable. The system considered in this paper is a generalization of those without delay or impulses that were studied in [2,12,13,19,21,22]. More precisely, we consider the following semilinear system: The paper is organized as follows. In Section 2, we recall some fundamental concepts and establish existence of mild solutions for system (1.1). In section 3, we present some criteria for the approximate controllability of system (1.1) in terms of the system defined by the linear part. Finally, in Section 4, an example is presented which illustrates the main theorem.

Preliminaries
In this section, we review some basic concepts, notations and properties needed to establish our results.
Definition 2.1. (see [23,24]) A one parameter family {C(t) : t ∈ R}, of bounded linear operators in the Banach space X is called a strongly continuous cosine family iff (ii) C(0) = I; (iiii) C(t)x is strongly continuous in t on R for each fixed x ∈ X.
Throughout this paper, A is the infinitesimal generator of a strongly continuous cosine family, {C(t) : t ∈ R}, of bounded linear operators defined on a Banach space X endowed with a norm · . We denote by {S(t) : t ∈ R} the sine function associated to {C(t) : t ∈ R} which is defined by Moreover, M and N are positive constants such that C(t) ≤ M and S(t) ≤ N for every t ∈ J.
The infinitesimal generator of a strongly continuous cosine family {C(t) : t ∈ R} is the operator A : X → X defined by where D(A) = {x ∈ X : C(t)x is twice continuously differentiable in t}, endowed with the norm Define E = {x ∈ X : C(t)x is once continuously differentiable in t}, endowed with the norm then E is a Banach space. The operator-valued function is a strongly continuous group of bounded linear operators on the space E × X generated by the operator A = follows that AS(t) : E → X is a bounded linear operator and that AS(t)x → 0 as t → 0 for each x ∈ E. Furthermore, [25] S(t + s) = C(t)S(s) + C(s)S(t), (2.1) 3) The following properties are well known [23]: The existence of solutions of the second order abstract Cauchy problem, where h : J → X is an integral function, has been discussed in [23]. Similarly, the existence of solutions of semilinear second order abstract cauchy problems has been treated in [24]. We only mention here that the function x(·) given by is called a mild solution of (2.4), and that when ς 1 ∈ E the function x(·) is continuously differentiable and where · is any norm of X.
Throughout, we set t 0 = 0, t m+1 = b, and for u ∈ PC we denote by u k , for A normalized piecewise continuous function x : [σ, τ ] → X is said to be normalized piecewise smooth on [σ, τ ] if x is continuously differentiable except on a finite set S, the left derivative exists on (σ, τ ] and the right derivative exists on [σ, τ ). In this case, we present by x (t) the left derivative at t ∈ (σ, τ ] and by x (σ) the right derivative at σ. We denote by PC 1 ([σ, τ ], X) the space of normalized piecewise smooth functions from [σ, τ ] into X and by PC 1 the space formed by all normalized piecewise smooth functions x : J → X such that S = {t k : k = 1, · · · , m}. Obviously, PC 1 is also a Banach space with the norm x PC 1 = max{ x PC , x PC }.
In this paper we employ an axiomatic definition of the phase space B which is similar to that introduced by Hale and Kato [10] and appropriated to treat retarded impulsive differential equations. Specifically, B is a linear space of functions mapping (−∞, 0] into X endowed with a seminorm · B . We assume that B satisfies the following axioms: (B) The space B is complete.
Example 2.1. (The phase space PC r × L p (ρ, X)). Let r ≥ 0, 1 ≤ p < ∞ and let ρ : (−∞, −r] → R be a non-negative measurable function. Assume that ρ satisfy the following conditions: is a set with Lebesgue measure zero. In order to define the solution of the system (1.1), we consider the space to be the seminorm in B h1 and B h2 defined by Motivated by the formula (2.5), we give the mild solution for the problem (1.1).

Definition 2.2.
A function x(·, φ, ϕ, u) ∈ B h2 is said to be a mild solution of (1.1) if m; (iv) x(·)| J ∈ PC 1 and the following integral equation is verified: (2.7) To establish our results, we introduce the following assumptions on system (1.1): is the controllability Grammian.
(H 2 ) f : J × B × B → X is a continuous function and there exist positive constants k 1 and k 2 such that for every ω 1 , ω 2 , ν 1 and ν 2 ∈ B.
(H 3 ) The functions I j k : X → X are continuous and there exist positive constants L(I j k ), j = 1, 2 such that For φ ∈ B, we define φ by and then φ ∈ B h1 . For ϕ ∈ B, we define ϕ by and then ϕ ∈ B h1 .
It is easy to see that x satisfies if and only if x satisfies x 0 = 0 and It is also easy to see that x satisfies if and only if x satisfies x 0 = 0 and For any x ∈ B h1 , It is now shown, ( x, z) ∈ Z implies x ∈ B h2 .
On the space Z, we define the nonlinear operator (2.9) The continuity and well-definedness of Φ follow directly from the assumptions. It will be shown that the operator Φ has a fixed point. (2.10) Similarly, we have 11) where N = sup t∈J AS(t) L(E,X) .
Next, we show that Φ is a contraction mapping on Q. For this, let us take ( x, z), ( v, w) ∈ Q, then we get In view of We have Finally, we have The above inequalities (2.15) and (2.16) and the assumption max{φ 1 , φ 2 } < 1 imply that Φ is a contraction mapping. Hence there exists a unique fixed point ( x, z) ∈ Q. Then the function x(·) = x(·) + φ(·) ∈ B h2 is a mild solution of (1.1). This completes the proof.

Approximate Controllability
In this section, we compare approximate controllability of the semilinear system (1.1) with approximate controllability of the associated linear system. For this reason, we consider the linear system with initial condition Both the exact and the approximate controllability of systems (3.1)-(3.2) have been studied by several authors. Directly related to systems modeled by (3.1)-(3.2), we mention the works [6,7,9,11,17,18,20,27,29,30]. The following result has been established by Fattorini [7] and Triggiani [27,28].
We introduce the sets where T (t) is the analytic semigroup generated by A [1,8]. It is clear that U 0 ⊆ U ∞ .
(c) If BU 0 is dense in BU and systems (3.1)-(3.2) are approximately controllable on J, then Sp{A n BU 0 : n ≥ 0} is dense in X.
We return to the controllability problem for the semilinear system (1.1). Before stating and proving our main result, we give first the definition of approximate controllability.
In a similiar way Using again that f is bounded, we infer that b bn C(b−s)f (s, x s , x s )ds → 0, n → ∞. Again, as in x(b, φ, ϕ, u n ), from (2.1) and (2.2), all the summation terms cancel as n → ∞. Thus, y(b, φ, ϕ, u n ) → z 2 as n → ∞.
This implies that z ∈ R(f, φ, ϕ). Because z was arbitrarily chosen, this completes the proof.

Example
In this section we present an example of controllable impulsive partial differential equation with infinite delay. In the following, X = L 2 ([0, π]); B = PC 0 × L 2 (ρ, X) and A : D(A) ⊆ X → X is the map defined by Af = f with domain D(A) = {f ∈ X : f and f are absolutely continuous, f ∈ X, f (0) = f (π) = 0}. It is well known that A is the infinitesimal generator of a strongly continuous cosine family of operators, {C(t) : t ∈ R} on X. Furthermore, A has a discrete spectrum and the eigenvalues are −n 2 , n ∈ N, with corresponding normalized eigenvectors z n (ξ) = 2 π sin(nξ); the set {z n , n ∈ N } is an orthonormal basis of X and the following properties hold: cos(nt) f, z n z n and S(t)f = ∞ n=1 sin(nt) n f, z n z n .
We define B : R → X by Bu = qu. Then B ≤ K = ∞ n=1 e −2n 2 q 2 n . To study the approximate controllability of (4.1), assume c(t) is measurable and continuous with Defining the operator f : J × B × B → X by Also defining the maps I 1 k and I 2 k I 1 k (w)(x) = π 0 K 1 (t k , x, y)w(y)dy, w ∈ X, x, y)w(y)dy, w ∈ X, then system (4.1) can be modelled as (1.1). Define We claim that It follows from Theorem 3.1 that the linear systems (3.1)-(3.2) are approximately controllable on J. Then the operator α(αI + Γ b t ) −1 → 0 in the strong operator topology as α → 0 + (see [4,27,28]). So assumption (H 1 ) is satisfied.

Conclusion
In this paper, the issue on the approximate controllability criteria for a class of second order impulsive functional differential systems with infinite delay has been addressed for the first time. A new set of sufficient conditions for the approximate controllability of the considered nonlinear systems have been established by using strongly continuous cosine families of operators and the contraction mapping principle. Particularly, we have shown that under the assumption that the approximate controllability of its linear part, this system is approximate controllable. Moreover, the example presented in Section 4 illustrated exactly an application of the obtained results. The neutral functional differential system in the form of x 0 = ϕ ∈ B, x (0) = w ∈ X, which was studied in [13], is a special case of our system. So our results are applicable to such system.